# Recent results on the spectra of lens spaces

## TL;DR

This paper reviews recent advances in understanding the spectra of lens spaces and related spaces, highlighting new methods, results, and open problems in spectral geometry.

## Contribution

It introduces a new elementary proof for the spectrum on functions and discusses recent developments, open problems, and conjectures in the spectral theory of lens spaces.

## Key findings

- Construction of isospectral manifolds that are not strongly isospectral
- New elementary proof avoiding representation theory for the spectrum on functions
- Recent results and open problems in spectral geometry of lens spaces

## Abstract

In this paper we report on recent results by several authors, on the spectral theory of lens spaces and orbifolds and similar locally symmetric spaces of rank one. Most of these results are related to those obtained by the authors in [IMRN (2016), 1054--1089], where the spectra of lens spaces were described in terms of the one-norm spectrum of a naturally associated congruence lattice. As a consequence, the first examples of Riemannian manifolds isospectral on $p$-forms for all $p$ but not strongly isospectral were constructed. We also give a new elementary proof in the case of the spectrum on functions. In this proof, representation theory of compact Lie groups is avoided and replaced by the use of Molien's formula and a manipulation of the one-norm generating function associated to a congruence lattice. In the last four sections we present several recent results, open problems and conjectures on the subject.